Question

The locus of the vertices of the family of parabolas

$y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is

• A. $xy=\frac{64}{105}$
• B. $xy=\frac{105}{64}$
• C. $xy=\frac{3}{4}$
• D. $xy=\frac{35}{16}$

Answer 1

The correct option is B $xy=\frac{105}{64}$

Convert the given equation to the standard form.

The equation of the parabola can be written as

$\frac{y}{a}=(\frac{ax}{\sqrt{3}}+\frac{\sqrt{3}}{4}{)}^2-\frac{3}{16}-2$

$(x+\frac{3}{4a}{)}^2=\frac{a^2}{3a}(y+\frac{35}{16}a)$

Vertex is $x=\frac{-3}{4a},y=\frac{-35a}{16}$

Therefore,

Locus of the vertex is $xy=\frac{105}{64}$.